Triangle solver



E. B. HALEs TBIANGLE soLvER Sept. 16, 1952 2 Simms- SHEET 1 .filed Jari. 21. 194s Snnentor EVERETT B. HALES i Gttomeg Sept. 16,I 1952 y E. B, HALES y 2,610,789l

' TRIANGLE soLvER Fild Jan. 21, 1948-- 2 SHEETS-SHEET 2 PHASE DETECTOR l mnentor EVERETT B.4 HALES (Ittorneg Patented Sept. 16, 1952 TRIANGLE SOLVER Everett B. Hales,pMount Kisco, N. Y., assignorto General. Precision Laboratory Incorporated, a

corporation of New York 4' Application January 21, 1948, Serial No. 3,652

, i 1 This invention relates to. al triangle solver.. More specifically the invention comprises a'system. in which twoalternating current voltages of equal frequencies but different amplitudes, selected asrepresentative of rectangular coordinates, that is, amplitudes in X and Y directions arecombined to produce asingle voltage Whose amplitude is proportional to the vector sum` of the amplitudes of the selected voltages and whose phase i'sproportional to the phase angle of that vectorl sum. :In other Words,y the system of the present inventionconverts voltages representative of-r rectangular coordinates to a singlevoltage of corresponding polar coordinates; In various trainingdevices, target simulators andthe like it is desired 'to indicate synthetically the continuously varying positions of a .number of targets or other .objects with respect to a selected reference point' ,which may also vary in position. This requires that a number of" controls be provided which may be so actuated that the position .ofv all ofthe targets may be continuouslyv varied in accordance "with some preconceived plan. Such continuous variations in position are most conveniently resolved into components of motion in'north-s'outh and east-West directions since by 'suchl're'solution changes in position may be accomplished by mere addition or subtraction of components of motion in these directions rather than requiring complicated vector sol'ution'slas would be thecase if thetargets were located and followed by changes of range and azimuth.

Itis simpler, therefore, to base the operation of the controlling means on a system of rectan-` gular coordinates and "to provide sets of controlsvv which may be actuated to produce voltages whose amplitudes are representative of the positions of the targets in X and Ydirections. When,.how -Y ever, such voltages are t'obe utilized on'a plan. position indicator typeof display, the rectangular coordinate voltages must first be converted to polar coordinate voltages. It isa function of the present invention to provide asimpleI yet acouatemeansforperformingthisconversion.

Various systems for accomplishing thisp'urpose haya been. devisedfin the'past `buttliey suffer' from the disadvantage that when a plurality of such*4 devices are connectedto. a singlesource of energy to simulate a plurality of targets, actuationof any" one'` such device tends to imbalance thev system and afiect-the-settings of the remainder.

*Inthe' system of the instant invention couver-- sion from: rectangular to polar coordinates is accomplished? Aby relatively simple and inexpensive components and?. numbers of1 suchsysterr'xsv may be operated from asingle source of energy Without the objectionable interaction hitherto obtained. While as indicated'above, `the invention is particularly adapted for use-in training devices'and target simulators, it will also be appreciated tha-t; the simple system of the present inventionfindsnumerous applications wherever it is desired to convert rectangular coordinates toV polar coordinates and to obtain an indication of the-magnitude and angle ofthe hypotenuse of a right tri, angle knowing; the magnitude of the adjacent sides or to vectoriallyadd voltages representative ci values in X-andYdirections. n l 1 A.

The invention will be more readily understood. from the following description considered inconnection with the attached drawings, in which:

Figure 1 is a simplified schematic diagram illus-- trative oi the system of the inventionl Figure 2 is a ,diagram illustrating the vector relationship of the various voltages applied to the circuit of Fig..l. Y Figure 3 is a schematic diagram f the circuit of Fig. 1 having in addition, means by which separatey outputs proportional to phase .and mag-- ntude may be obtained. I-

Referring particularly to `Figi1.alternating current energy isvgenerated by an oscillator II which may be operated at any desired and con-1 The vvoltages Which are chosen-tothe" repr# sentative of voltages in X and Y directions are obtained from potentiometers Il and I8 connected in parallel to the' voltage supply circuit I2. Movable contacts I9 and 2l may be adjusted throughout the range 'of these' potentiometers so that the voltages impressed on conductors 22 and 23 may be varied from a, maximum of one phase to a maximum of the opposite phase.

The voltage existing between. conductors' -22 and 23 isA therefore representative of the vector sum of the voltages. impressed. on. these conductors by manipulation of the movable contacts I9 and 2|. Referring to the diagram of Fig. 2 and considering ther voltage impressed oncontactzl,

and hence conductorr 23 as represented by the vector X and the v'oltage'ihipressed on contact I 9 and hence the conductor 22 as represented by the vector Y, the voltage existing between the two conductors will be representative of the vector sum of these voltages having an amplitude E and a phase angle 0.

The conductors 22 and 23 are connected tofopposite ends of a series network consisting of a resistance 24 and a condenser 25 and a Voltage Eo obtained between a conductor 26 connected to the juncture of resistance 24 and condenser 25 and a conductor 2T connected to ground. By proper adjustment and selection of values for the resistance 24 and capacity 25 in relation to each other and the frequency of the voltage supply source I l, the voltage Eo can be made to have an amplitude which is proportional to the voltage E, i. e., the vector sum of the selected voltages X and Y and to have a phase proportional to the phase angle of the vector sum of voltages X and Y. That is to say, a voltage Eo is obtained whose polar coordinates are representative of the rectangular coordinates of the voltages X and Y and by proper calibration the desired conversion from rectangular to polar coordinates is realized.

The manner in which this conversion is accomplished, the values which must necessarily be imparted to the circuit components and the iixed relationship of the voltage Eo to the Vector sum of voltages X and Y, i. e., the voltage E may be mathematically expressed as follows:

The voltage Eo which is the voltage oi conductor 26 as respects ground is equal to the algebraic sum of the voltage Y and the Voltage existing across condenser 25. The voltage existing across condenser 25 is in turn a function of the potential' existing between conductors 22 and 23, namely, voltages X and Y and the reactance of resistance 24 and condenser 25, The voltage Eo may, therefore, be expressed by the following expression:

1 jC R+!- where En, Y and X have the values as above, Ris the resistance of resistance 24, C the capacity of condenser 25, e the frequency of the supply source in radians per second, i. e., w=21rf and 7' has the value of \/-1 as used in complex notations.

Multiplying the numerator and denominator of the fraction of Equation 1 by the value ywC Equation 1 may be reduced to the form:

Selecting values of resistance and capacity such that wCR=l and substituting in Equation 2 it follows that:

Multiplying the numerator and denominator by the value (1-9' 1) Equation 4 becomes:

which may be expanded to:

E=\/X2+2XY+ Y2+ 1v1-anhu;2 (s) which when like terms are combined simplifies to: EO=J2 X2+Y2 #ga/xavi 9) From the geometry of Fig. 2 it is apparent that:

E=\/X2+ Y2 (l0) E being the vector sum of X and Y.

Substituting the value E as determined by Equation 10 for its equivalent \/X2{Y2 in Equation 9 it follows that L E0 Vi .707E (ll) It is apparent therefore that the amplitude of the voltage Eo is directly proportional to themag'- nitude of the voltage E, that is, the vector' sum of voltages X and Y.

Considering now the relationship ofthe phase angles of the voltage Eo and the vector E of Fig. 2 and denoting the phase angle of Eo by the notation e it is apparent from the vector relationship of Equation 6 that:

lit YH? X -i-Y being the real term of Equation 6 and hence the adjacent side Y-X is the imaginary term and hence the opposite side of the vector triangle Whose included angle is a.

Dividing numerator and denominator by the jvalue X, Equation 12 may be converted to the crm:

tan= (12) From the geometry of Fig. 2 it is readily appar-4 ent that the phase vangle of the vector E, namely 0, may be expressed by the equation:

tan o:

tan 6:;

hence substituting the' value as determined by Equation 14 in Equation 13 there is obtained the relationship: y f

tan 0- 1 tan @+1 By simple trigonometric conversion tan determined by Equation 15 becomes:

=tan il (15) anlassepolar. aioordinatesslnnut Vvoltages,` expressed n frectangu1arfcoOrdin'ates..y ctherwordasemcted input voltages whose amplitudes represent; the two vectors displaced 90 with respect to each other may be combined in a single phase shifting network to'fyield an'output whose magnitude istrepresentati've of,F theV vector sum4 of the input voltages and whose phase isqrepresentative of the phaseangleof suchvector sum.y

Likewise,` lsince the reactance of the network consistingof resistance 24-and condenser 425may be` nfxade.A higha number of icon-version networks Zmay be` connected tOcperatei-'rom'a singie source 'throug-lia series caf-potentiometers similar to those of I1 and |31 as depictedfin Fig. 1 without aifectng the balance of the supply system at any adjustment ofthe input voltages. 'That is' to isay, since the' current flowing through thev network consisting of resistance 215`V and condenser t'may be made negligible even lwhen the contacts I9 and 2| arel adjusted to extreme positions in their ranges insuiiicient current is drawn by the network 24 and 25 to aiTect the balance of the two halves of secondary |4 and hence the operation of any duplicate networks which may be con# nected to the same source of supply through potentiometers similar to I1 and I8.

As an example only, of values which have been found to give desirable results a resistance of approximately 230,000 ohms and a condenser having a capacity of v680 micromicrofarads have been found to operate satisfactorily when supplied by 1000 cycle alternating current source.

Similar results may be obtained in a circuit wherein an inductance is substituted for the condenser 25. In a practical circuit, however, itis more advisable to use a condenser since it is diii-v cult to obtain an inductance of suitable value which at the same time has negligible resistance.

In utilizing a voltage such as produced by the system of Fig. 1 in which two variables are expressed in phase and amplitude, it may be desirable to provide separate outputs one of which is proportional to the phase variation and the other of which is proportional to the amplitude variation.

The schematic diagram of Fig. 3 illustrates one means by which this separation of variables may be obtained.

The X and Y voltages representative of rectangular coordinates are obtained by use of potentiometers 3| rand. 32 connected to an alternating'supply source of the lsame nature as illustrated in Fig. 1. The selected potentials derived from potentiometers 3| and 32 by means of movable contacts 33 and 34 are applied to the opposite ends of a phase shifting network similar t the phase shifting network 24 and 25 of Fig. l except that in addition to a iixed resistance 36 and condenser 31 and additional adjustable resistance 38 is included in this series circuit so that these elements may be accurately adjusted to satisfy the requirement wC'R=1.

The juncture of Aresistance 36 and condenser 31 is connected to the grid of a cathode follower 40 the cathode of which is connected to the negative side of a power supply through a relatively large unbypassed cathode resistance 39 and hence the voltage developed by the network 36, 31 and 38 which is of varying phase and amplitude as heretofore described is applied to the input of the cathode follower 49.

The output of this cathode follower is connected to the primary 4| of a step-up transformer 42. Included in the output of the cathode follower-:MIIA and;l 'iny series with the primary, '4 I,

a phase` shifting` network' is provided. consisting of. condenserA andi potentiometer S44; Values are so chosen; :that adjustment of potentiometer 444 will' provides; phase. adjustment. offrom Oto l0!" across; condenser .43 and; by-fthis" means' the slight differences in phase shi-ft of' thevarious .tran'sformersmay be: compensatedV by `suitable adjustmentof potentiometer44. Y Thesecondary 465 of transformer 42 is connectedto the inputf of `a second cathode follower 4-1 Ithrough the medium ota ypotentiometer 48 con stitutingan amplitudeadjusting means. V

-.When the potentiometer 44isadjusted to ati-'- lust the phase and! sovcompensate for the phase shi'ftof the transformers in circuit such adjust'- 'menti also changes the amplitude and therefore potentiometer 48 must. be adjusted to compensate forthechange in amplitude resulting Afrom adjustment-oi? the-phase shift'. Relatively large changes in adjustment of potentiometer 48 will result in some change in phase so that a second adjustment of potentiometer 44 followed by a second adjustment of potentiometer 48 will frequently be necessary. However, as a small adjustment of potentiometer 48 causes no appreciable change in phase it will rarely be necessary to repeat this process a second time.

The grid of the cathode follower 41 is connected to the negative terminal of the power supply source through a potentiometer 49 which permits proper zero adjustment of the system. That is, the potentiometer 49 may be so adjusted that at an input of zero amplitude, the output is likewise Zero.

The output of cathode follower 41 is impressed on a phase detector 4which may also include a constant current amplifier and the output derived from this circuit by the conductor 52 contains the information as to phase of the voltage derived from the network 35, 31 and 38.

The signal derived from the potentiometer 48 is also applied to the grid of a peak rectifier 53 through an isolating resistance 54 and the output of this rectiier includes a filter network consisting of condensers 56 and 51 and resistances 5B and 59 so that a direct current voltage is obtained whose value is representative of the amplitude of the voltage applied to the input of the cathode follower 40.

This potential is applied to the input of a cathode follower 6| and the output thereof impressed on conductor 62 contains the information as to amplitude of the voltage derived vfrom the network 36, 31 and 38 thus two separate outputs are provided, one containing phase information and the other containing amplitude information.

What is claimed is:

1. In a system of the character described, an alternating current circuit including a tra-nsformer, the secondary of which is grounded at its midpoint, rst and second potentiometers connected in shunt to said alternating current circuit, an independently movable contact for each of said potentiometers, a resistor and a condenser connected in series between said movable contacts, said resistor and condenser having relative values of resistance and capacity as determined by the expression wCR=1 where C is the capacity of the condenser, R the resistance of the resistor and w the frequency of the alternating current circuit in radians, and an output circuit connected between the juncture of said resistor and condenser and ground.

7 '2. Ina system of the character described,"an alternating current source, a circuit connected to said source having its midpoint grounded,` a pair oi` potentiometers connected across said circuit each of said potentiometers including an independently movable contact, a resistor and a condenser connected in series between said movable contacts, said resistor and condenser having such values that the capacitive reactance of said condenser is equal to the resistance of said resistor at the frequency of said alternating current source, an output circuit connected between the juncture of said resistor'and condenser and ground, said output circuit including a phase de- .tector producing as an output thereof a constant amplitudealternating potential whose phase is .proportional to the phase of the quadrature vector sum of the potentials impressed on said movable contacts, and a'peak rectiiier producing as an .output a direct current potential, the magnitude 8 of which is proportional tothe magnitude of-.the quadrature vector sum of the potentials impressed on said movable contacts. EVERETT B. HALES.

. REFERENCE S CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS OTHER REFERENCES Radio Engineers Handbook, Terman, published 1943, McGraw-Hill Co., New York, N. Y., 1019 pages (page 949, Figure 56a). 

